$C^{1,1}$ regularity of geodesics of singular K\"{a}hler metrics
Differential Geometry
2021-01-20 v2 Analysis of PDEs
Complex Variables
Abstract
We show the optimal regularity of geodesics in nef and big cohomology class on K\"ahler manifolds away from the non-K\"ahler locus, assuming sufficiently regular initial data. As a special case, we prove the regularity of geodesics of K\"ahler metrics on compact K\"ahler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge-Amp\`ere equation that does not require strict positivity of the reference form near the boundary. We also discuss the case of some special geodesic rays.
Keywords
Cite
@article{arxiv.1901.02105,
title = {$C^{1,1}$ regularity of geodesics of singular K\"{a}hler metrics},
author = {Jianchun Chu and Nicholas McCleerey},
journal= {arXiv preprint arXiv:1901.02105},
year = {2021}
}
Comments
37 pages; Corrected an incorrect formula in the proof of the main result; Results are unchanged; Exposition is improved and more background material is added; Final version, to appear in J. Lond. Math. Soc